Question

$$ \frac { ( x + 3 ) ^ { 2 } } { 9 } + \frac { ( y + 1 ) ^ { 2 } } { 16 } = 1 $$ $$ \frac { ( x + 3 ) ^ { 2 } } { 16 } + \frac { ( y - 1 ) ^ { 2 } } { 9 } = 1 $$ $$ \frac { ( x - 3 ) ^ { 2 } } { 16 } + \frac { ( y + 1 ) ^ { 2 } } { 9 } = 1 $$ $$ \frac { ( x - 3 ) ^ { 2 } } { 9 } + \frac { ( y - 1 ) ^ { 2 } } { 16 } = 1 $$

Explanation:

To determine the correct ellipse equation, we analyze the center, major/minor axes from the graph (let's assume we're matching to graph b, for example, but let's go through the standard ellipse form:

The standard form of an ellipse is \(\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1\) (vertical major axis, \(a > b\)) or \(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\) (horizontal major axis, \(a > b\)), where \((h, k)\) is the center.

Step 1: Identify the center from the graph

For graph b (left - shifted ellipse), the center appears to be \((-3, 1)\)? Wait, no—wait, let's re - examine. Wait, maybe graph b has center \((-3, 1)\)? Wait, no, let's check the equations:

Wait, let's take the second equation: \(\frac{(x + 3)^2}{16} + \frac{(y - 1)^2}{9} = 1\). Let's break it down:

• Center: \((h, k)\) from \((x - h)^2\) and \((y - k)^2\). So \(x+3=(x - (-3))\), so \(h=-3\); \(y - 1\) means \(k = 1\). So center \((-3,1)\).
• Denominators: \(16\) (under \(x\) - term) and \(9\) (under \(y\) - term). Since \(16>9\), the major axis is horizontal (along the \(x\) - axis), with \(a = 4\) (since \(a^2=16\)), \(b = 3\) (since \(b^2 = 9\)).

Now, let's check the graph (graph b): it's a left - shifted ellipse, center around \((-3,1)\), horizontal major axis (wider along \(x\) - axis). Let's verify other equations:

1. \(\frac{(x + 3)^2}{9}+\frac{(y + 1)^2}{16}=1\): Center \((-3,-1)\), vertical major axis (since \(16>9\))—doesn't match graph b.
2. \(\frac{(x + 3)^2}{16}+\frac{(y - 1)^2}{9}=1\): Center \((-3,1)\), horizontal major axis (\(a = 4\), \(b = 3\))—matches the left - shifted, horizontal ellipse in graph b.
3. \(\frac{(x - 3)^2}{16}+\frac{(y + 1)^2}{9}=1\): Center \((3,-1)\)—right - shifted, doesn't match.
4. \(\frac{(x - 3)^2}{9}+\frac{(y - 1)^2}{16}=1\): Center \((3,1)\), vertical major axis—doesn't match.

So the correct equation is \(\boldsymbol{\frac{(x + 3)^2}{16}+\frac{(y - 1)^2}{9}=1}\) (the second option).

Answer:

\(\frac{(x + 3)^2}{16}+\frac{(y - 1)^2}{9}=1\) (the second equation in the list: \(\frac{(x + 3)^2}{16}+\frac{(y - 1)^2}{9}=1\))